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Mathematics

MATH 215. Abstract Algebra, spring 2011
Ben Newton, newtonb@beloit.edu

Student Commitment: minimal
Developmental Stage: basic
Community partner(s): Help Yourself, coordinated by the LAPC
Resources: Course Syllabus (pdf), Help Yourself Project Guidelines (pdf)

Goals:

  • To learn about groups, rings, and fields, and the basics of how they are studied.
  • To develop skills in abstracting the general principles underlying specific mathematical operations and structures.
  • To become proficient at writing proofs, and more generally, at discussing math in a clear and precise manner.

Courses and the Community: Teaming up with Help Yourself

by Ben Newton, Professor of Mathematics and Computer Science

The Help Yourself Program at Beloit College has organized tutoring and college preparation sessions, as well as weekend workshops for local high school and middle school students since 1986. The program has a fantastic track record, with a 100% high school graduation rate, and 98% of participants in recent years going on to attend a 4-year college immediately upon graduation. Many Beloit College faculty and students have been involved in the program in various capacities over the years, but until recently this involvement has not typically been connected to the college curriculum.

This spring I was very fortunate to be able to take part in a pilot project that sought to link several of the weekend workshops with existing Beloit College courses. The project was organized by Carol Wickersham, and besides myself, the participating faculty were Scott Espeseth (Art), Katie Johnson (Biology), and Jingjing Lou (Education).

The selected courses were ones that were already scheduled to be taught in the spring semester. Students in each of these courses were put in charge of the design and implementation of one Saturday workshop for about 20-30 high school or middle school students. The workshops were to be based around the content of the courses that they were taking, and their work in this effort would essentially amount to a major course assignment, among their other readings, homework, exams, etc.

The course that I included as part of this project was Math 215: Abstract Algebra, which is typically taken by sophomore and junior math majors or junior and senior math minors. This spring, 16 students were enrolled. Students in the course were divided into three teams, each responsible for a separate activity which the Help Yourself students would rotate between during the workshop.

My expectation was that the activities would be designed with various mathematical concepts from our course in mind. On the back-end, I hoped that students would be able use the language of abstract algebra (i.e. technical terms such as ‘associativity’ and ‘isomorphism’) in describing their activities to me, but on the front-end, the goal was to create fun and challenging puzzles and contests that the Help Yourself students could fully engage in without having to learn numerous abstract definitions or master complicated techniques.

The teams showed remarkable creativity and resourcefulness in putting their activities together. One team used the theory of permutation groups to create a contest in which two teams of Help Yourself students raced to rearrange playing cards according to various rules. Another team created Sudoku-like puzzles to illustrate various properties of binary operations, and the third designed a wonderful collection of icons and abstract patterns which they put on cards for a game based on the concepts of sets and relations.

Overall, I’d have to say that our workshop on March 19 was quite successful. One of my goals for the Help Yourself students was to impart a sense of the usefulness of abstract thinking in recognizing and describing patterns, and to make them aware that mathematics is as much about these processes as it is about techniques for numerical computation. From my perspective, it seemed like there was at least some appreciation of this point, but I think that workshop also served an even more important purpose, which was to allow the Help Yourself students to interact with Beloit College students in a campus setting and in the context of one of their courses.

Of course, the day was not without its small hiccups. Even with preparation and rehearsal in a classroom setting, it proved difficult to fully envision what the activities would be like in practice. More than one team had a hard time, especially with their first group, getting on the same page as the Help Yourself Students about what the students were being asked to do. All three teams reported, however, that their last group of the day had the most success in completing the challenges put before them. This might be due to improvements in the teaching and explanations of the Math 215 students, or it might be due to the Help Yourself students starting to get a feel for the kinds of abstract thinking that they were being asked to do. Hopefully, both of these factors played a part.

Finally, a large majority of the Math 215 students said that they enjoyed taking part in the workshop, but there were mixed opinions about whether they would have chosen to spend the amount of time that we did in preparation for it. The in-class time that we spent was spread out over multiple days, but I’d estimate that it amounted to between two and three Tuesday/Thursday class periods in total. Several students indicated that they would have preferred to use this time in a way that was more directly focused on the content of the course.

A central conception behind the entire pilot project was that the efforts involved in preparing for the workshop could be an important part of the content learning of a given course, as opposed to being a separate but still valuable exercise. In the case of Math 215, I felt that this process could really solidify students’ knowledge of key ideas by having them reframe them in another context. In retrospect, it may be that some of the more advanced students in the class were already at this level of understanding before the preparations began, but I don’t think that this means that there is nothing to be gained for these students in terms of course content in such a situation. For example, the team designing the card-rearranging activity encountered the problem of determining which of the puzzles that they came up with actually had a solution. This turned out to be an interesting and thought-provoking problem that was just at the right level for our course.

I think that it is a very worthwhile goal for instructors taking part in this type of project in the future to find a way to structure assignments so that the project not only benefits students needing a little extra help in mastering core concepts, but also provides difficult challenges for students who are further along in their understanding.